By "inertial force" you mean the force pushing back on the accelerating object. This isn't "inertial force" it's called the "reaction force" and it appears whether the body is not moving or moving.
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Ron MaimonMay 9 '12 at 20:20

No i don't mean the reaction force. What i mean by inertial force is: If i was floating in space along with a satellite in rest; When i push the satellite with a force, the satellite resists to the movement proportional to its mass and its current acceleration.
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bijanMay 9 '12 at 23:32

And that current acceleration was created by my force.
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bijanMay 9 '12 at 23:34

meaning in non-inertial frame of reference you have to add pseudo or fictitious or inertial force $\vec{F'}$ if you wish to use $\vec{F} = m \cdot \vec{a}$.

$\vec{a'}$ in second expression represents the acceleration of the non-inertial frame of reference and not the acceleration of the mass. Also, note minus sign in the second expression, which means that pseudo force is directed in the opposite direction to acceleration of the frame of reference!

By the way, non-inertial frame of reference is usually conveniently chosen in a way that observed body is at rest (position of the body defines that frame of reference), so you end up with $\vec{a} = 0$.

Yes, they are. Plainly speaking (sorry Nick), the first formula is "defined" only within inertial frame of reference. But if you want to use the first also in non-intertial frame, you have to use the second too.
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PygmalionMay 8 '12 at 20:44

Actually, neither of those is quite correct. The first expression is subtly wrong; the second may be either subtly wrong in the same way, or completely wrong, depending on what you mean by "inertial force."

What Newton's second law in this form really means is that the net (total) force acting on an object at time $t$ is equal to the object's mass times the object's acceleration at that time. It does not apply to any one individual force, only to the net force. For this reason it is most properly written

$$\sum \vec{F} = m\vec{a}$$

with the summation symbol to indicate the sum over all forces.

There are some restrictions on the validity of this equation; in particular, as Pygmalion pointed out, it only works in an inertial frame of reference (or can be taken to define an inertial frame of reference, in Newtonian mechanics). Also, it only works for objects whose mass is constant, and whose velocity is small. (That last one is sort of debatable depending on which definition of force you use) The more general form of Newton's second law (and in fact the way it was originally written) is

That's fair. The reason I answered as I did was that I have heard "inertial force" used to represent something that is completely wrong (when I'm teaching students about forces they sometimes invent a force that is due to an object's mass and call it the "force of inertia") so I thought that was what the question mentioned.
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David Z♦May 9 '12 at 20:04

1

@DavidZaslavsky After additional brainstorming, checking books and Googling, I added "(net or total)" to my text, leaving the formula the same, as it is actually commonly used. So you are right on that one. However, I think net force applies only in the case of the original Newton's laws. When talking about pseudo forces, it has nothing to do with net force.
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PygmalionMay 9 '12 at 21:22

Newton meant both, since action equals reaction. The first is Newton's intended interpretation--- to accelerate a body by a, you need a force equal to ma. The second is the necessary reaction force on the thing that is doing the acceleration. If you push on a body to accelerate it, the body pushes back on you in the opposite direction.

Note that the reaction doesn't care if the body is speeding up or not--- whenever you apply F units of force on a body, the reaction is there, pushing you back by an equal amount.